I have VS 2008, 2010, and 2012 installed.
Initially, VS2013 Team Explorer was installed (Shell only). I uninstalled that.
Now, the Visual Studio Version Selector shows an empty list when executing a .sln file. Nothing shows.
How can I repopulate this list? Where is it stored? Registry? I tried to find entries, but since it doesn't actually have any items in the list, I couldn't search for a specific string.

I just encountered the same issue after installing Visual Studio 2015 parallel to an existing 2013 installation. In my case it turned out that that the problem was related to the solution file itself: It seems that the version selector does not like BOMs etc. (don't know how the solution got crippled, though). Make sure that the solution starts with
and has no space, non-printable character etc. before that. After saving the file, the effect should be immediately visible with the correct icon returning.